Part I Crystalline solids

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, there is an enormous amount of information about the world, if just a little imagination and thinking are applied. (R. P. Feynman, The Feynman Lectures on Physics)

Solids are the physical objects with which we come into contact continuously in our everyday life. Solids are composed of atoms. This was first postulated by the ancient Greek philosopher Demokritos, but was established scientifically in the 20th century. The atoms (ατoµα = indivisible units) that Demokritos conceived bear no resemblance to what we know today to be the basic units from which all solids and molecules are built. Nevertheless, this postulate is one of the greatest feats of the human intellect, especially since it was not motivated by direct experimental evidence but was the result of pure logical deduction.

There is an amazing degree of regularity in the structure of solids. Many solids are crystalline in nature, that is, the atoms are arranged in a regular three-dimensional periodic pattern. There is a wide variety of crystal structures formed by different elements and by different combinations of elements. However, the mere fact that a number of atoms of order 1024 (Avogadro’s number) in a solid of size 1 cm3 are arranged in essentially a perfect periodic array is quite extraordinary. In some cases it has taken geological times and pressures to form certain crystalline solids, such as diamonds. Consisting of carbon and found in mines, diamonds represent the hardest substance known, but, surprisingly, they do not represent the ground state equilibrium structure of this element. In many other cases, near perfect macroscopic crystals can be formed by simply melting and then slowly cooling a substance in the laboratory. There are also many ordinary solids we encounter in everyday life 1 2 Part I Crystalline solids in which there exists a surprising degree of crystallinity. For example, a bar of soap, a chocolate bar, candles, sugar or salt grains, even bones in the human body, are composed of crystallites of sizes between 0.5 and 50 µm. In these examples, what determines the properties of the material is not so much the structure of individual crystallites but their relative orientation and the structure of boundaries between them. Even in this case, however, the nature of a boundary between two crystallites is ultimately dictated by the structure of the crystal grains on either side of it, as we discuss in chapter 11.

The existence of crystals has provided a tremendous boost to the study of solids, since a crystalline solid can be analyzed by considering what happens in a single unit of the crystal (referred to as the unit cell), which is then repeated periodically in all three dimensions to form the idealized perfect and infinite solid. The unit cell contains typically one or a few atoms, which are called the basis. The points in space that are equivalent by translations form the so called Bravais lattice . The Bravais lattice and the basis associated with each unit cell determine the crystal. This regularity has made it possible to develop powerful analytical tools and to use clever experimental techniques to study the properties of solids.

Real solids obviously do not extend to infinity in all three dimensions – they terminate on surfaces, which in a sense represent two-dimensional defects in the perfect crystalline structure. For all practical purposes the surfaces constitute a very small perturbation in typical solids, since the ratio of atoms on the surface to atoms in the bulk is typically 1 : 108. The idealized picture of atoms in the bulk behaving as if they belonged to an infinite periodic solid, is therefore a reasonable one. In fact, even very high quality crystals contain plenty of one-dimensional or zero-dimensional defects in their bulk as well.

It is actually the presence of such defects that renders solids useful, because the manipulation of defects makes it possible to alter the properties of the ideal crystal, which in perfect form would have a much more limited range of properties. Nevertheless, these defects exist in relatively small concentrations in the host crystal, and as such can be studied in a perturbative manner, with the ideal crystal being the base or, in a terminology that physicists often use, the “vacuum” state . If solids lacked any degree of order in their structure, study of them would be much more complicated. There are many solids that are not crystalline, with some famous examples being glasses, or amorphous semiconductors. Even in these solids, there exists a high degree of local order in their structure, often very reminiscent of the local arrangement of atoms in their crystalline counterparts. As a consequence, many of the notions advanced to describe disordered solids are extensions of, or use as a point of reference, ideas developed for crystalline solids. All this justifies the prominent role that the study of crystals plays in the study of solids. Part I Crystalline solids 3 It is a widely held belief that the crystalline state represents the ground state structure of solids, even though there is no theoretical proof of this statement. A collection of 1024 atoms has an immense number of almost equivalent ordered or disordered metastable states in which it can exist, but only one lowest energy crystalline state; and the atoms can find this state in relatively short time scales and with relatively very few mistakes! If one considers the fact that atomic motion is quite difficult and rare in the dense atomic packing characteristic of crystals, so that the atoms have little chance to correct an error in their placement, the existence of crystals becomes even more impressive.

The above discussion emphasizes how convenient it has proven for scientists that atoms like to form crystalline solids. Accordingly, we will use crystals as the basis for studying general concepts of bonding in solids, and we will devote the first part of the book to the study of the structure and properties of crystals.